How to sort file names in ascending order?

What you are asking for is numerical sort. You need implement a Comparator and pass it to the Arrays#sort method. In the compare method you need to extract the number from each filename an then compare the numbers.

The reason why you get the output you are getting now is that sorting happens alphanumerically

Here a is a very basic way of doing it. This code uses simple String-operation to extract the numbers. This works if you know the format of the filename, in your case Spectrum_<number>.txt. A better way of doing the extraction is to use regular expression.

public class FileNameNumericSort {

    private final static File[] files = {
        new File("Spectrum_1.txt"),
        new File("Spectrum_14.txt"),
        new File("Spectrum_2.txt"),
        new File("Spectrum_7.txt"),     
        new File("Spectrum_1000.txt"), 
        new File("Spectrum_999.txt"), 
        new File("Spectrum_9990.txt"), 
        new File("Spectrum_9991.txt"), 
    };

    @Test
    public void sortByNumber() {
        Arrays.sort(files, new Comparator<File>() {
            @Override
            public int compare(File o1, File o2) {
                int n1 = extractNumber(o1.getName());
                int n2 = extractNumber(o2.getName());
                return n1 - n2;
            }

            private int extractNumber(String name) {
                int i = 0;
                try {
                    int s = name.indexOf('_')+1;
                    int e = name.lastIndexOf('.');
                    String number = name.substring(s, e);
                    i = Integer.parseInt(number);
                } catch(Exception e) {
                    i = 0; // if filename does not match the format
                           // then default to 0
                }
                return i;
            }
        });

        for(File f : files) {
            System.out.println(f.getName());
        }
    }
}

Output

Spectrum_1.txt
Spectrum_2.txt
Spectrum_7.txt
Spectrum_14.txt
Spectrum_999.txt
Spectrum_1000.txt
Spectrum_9990.txt
Spectrum_9991.txt

The currently accepted answer does this only for numeric suffixes of files that are always called the same name (i.e. ignoring the prefix).

A much more generic solution, which I blogged about here, works with any file name, splitting names in segments and ordering the segments numerically (if both segments are numbers) or lexicographically, otherwise. Idea inspired from this answer:

public final class FilenameComparator implements Comparator<String> {
    private static final Pattern NUMBERS = 
        Pattern.compile("(?<=\\D)(?=\\d)|(?<=\\d)(?=\\D)");
    @Override public final int compare(String o1, String o2) {
        // Optional "NULLS LAST" semantics:
        if (o1 == null || o2 == null)
            return o1 == null ? o2 == null ? 0 : -1 : 1;

        // Splitting both input strings by the above patterns
        String[] split1 = NUMBERS.split(o1);
        String[] split2 = NUMBERS.split(o2);
        for (int i = 0; i < Math.min(split1.length, split2.length); i++) {
            char c1 = split1[i].charAt(0);
            char c2 = split2[i].charAt(0);
            int cmp = 0;

            // If both segments start with a digit, sort them numerically using 
            // BigInteger to stay safe
            if (c1 >= '0' && c1 <= '9' && c2 >= '0' && c2 <= '9')
                cmp = new BigInteger(split1[i]).compareTo(new BigInteger(split2[i]));

            // If we haven't sorted numerically before, or if numeric sorting yielded 
            // equality (e.g 007 and 7) then sort lexicographically
            if (cmp == 0)
                cmp = split1[i].compareTo(split2[i]);

            // Abort once some prefix has unequal ordering
            if (cmp != 0)
                return cmp;
        }

        // If we reach this, then both strings have equally ordered prefixes, but 
        // maybe one string is longer than the other (i.e. has more segments)
        return split1.length - split2.length;
    }
}

This can also handle version with subversions, e.g. things like version-1.2.3.txt